The electroweak oblique parameters, which parameterize new physics effects for the gauge boson self-energies, were first proposed in Ref. [1]}.
Their analytical expressions in the multi-Higgs doublet models were calculated in Refs. [2]}, [3]}.
With the definitions of \(S\)  and \(T\)  parameters given in Ref. [4]}, their analytical expressions in the 3HDM with axion can be derived from the new scalar boson loop contributions to the gauge boson two point functions.
We give the general formulae for the \(S\)  and \(T\)  parameter in Appendix , which are used in the numerical calculations in Sec.  and .
Since the the general formulae for the \(S\)  and \(T\)  parameters are somewhat lengthy,
we here describe the analytical expressions in the alignment limit.
New physics contributions to the \(S\)  and \(T\)  parameters, \(\Delta S=S_{\rm 3HDM}-S_{\rm SM}\)  and \(\Delta T=T_{\rm 3HDM}-T_{\rm SM} \) , are written in terms of the Passarino-Veltman functions [5]} as
\(4\pi \Delta {S}&=s_{\alpha _3+{\gamma _1}}^2 \left\lbrace \Delta B_5(m_Z^2,m_{H_3},m_{A_1})+\Delta B_5(m_Z^2,m_{H_2},m_{A_2})\right\rbrace \\&+c_{\alpha _3+{\gamma _1}}^2 \left\lbrace \Delta B_5(m_Z^2,m_{H_2},m_{A_1})+\Delta B_5(m_Z^2,m_{H_3},m_{A_2})\right\rbrace \\&-\Delta B_5(m_Z^2,m_{H^\pm _1},m_{H^\pm _1})-\Delta B_5(m_Z^2,m_{H^\pm _2},m_{H^\pm _2})\;, \\\Delta {T}&=-\frac{G_f}{8\sqrt{2}\pi ^{2}\alpha _{em} } \Big [c_{{\gamma _1}-{\gamma _+}}^2 B_5(0,m_{A_1},m_{H^\pm _1})-c_{\alpha _3+{\gamma _1}}^2 B_5(0,m_{H_2},m_{A_1})\\&+s_{{\gamma _1}-{\gamma _+}}^2 B_5(0,m_{A_1},m_{H^\pm _2})-s_{\alpha _3+{\gamma _1}}^2 B_5(0,m_{H_3},m_{A_1})\\&+s_{{\gamma _1}-{\gamma _+}}^2 B_5(0,m_{A_2},m_{H^\pm _1})-s_{\alpha _3+{\gamma _1}}^2 B_5(0,m_{H_2},m_{A_2})\\&+c_{{\gamma _1}-{\gamma _+}}^2 B_5(0,m_{A_2},m_{H^\pm _2})-c_{\alpha _3+{\gamma _1}}^2 B_5(0,m_{H_3},m_{A_2})\\&+c_{\alpha _3+{\gamma _+}}^2 B_5(0,m_{H_2},m_{H^\pm _1})+s_{\alpha _3+{\gamma _+}}^2 B_5(0,m_{H_3},m_{H^\pm _1})\\&+s_{\alpha _3+{\gamma _+}}^2 B_5(0,m_{H_2},m_{H^\pm _2})+c_{\alpha _3+{\gamma _+}}^2 B_5(0,m_{H_3},m_{H^\pm _2})\Big ],\) 
